ChE 452 Lecture 21 Potential Energy Surfaces

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Last Time Collision Theory Assumes reactions occur whenever

reactants collide Key equations

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k = v 0 A BC A BCc

(7.26)

v 2.52 10 Åsec

T300K

1AMUABC

131/2

ABC

1/2

(7.29)

Preexponentials Really Used By The Same Order As Collision Theory?

3

Table 7.2 a selection of the preexponentials reported by Wesley [1980]Reaction Preexponential

Å3/molecule SecReaction Preexponential

Å3/molecule SecH+C2H6

C2H5+H2 1.6 1014 O+C2H6 OH+C2H5 2.5 1013

H+CH H2+C 1.1 1012 O+C3H8 (CH3)2CH+OH 1.4 1010

H+CH4 H2+CH3 1 1014 O2+H OH+O 1.5 1014

O+H2 OH+H 1.8 1013 OH+OH H2O+O 1 1013

O+OH O2+H 2.3 1013 OH+CH4 H2O+CH3 5 1013

O+CH4 CH3+OH 2.1 1013 OH+H2CO H2O+HCO 5 1013

O+CH3 H+CH3O 5 1013 OH+CH3

H+CH3O 1 1013

O+HCO H+CO2 5 1012 OH+CH3 H2O+CH2 1 1013

Comparisons Between Collision Theory And Experiments

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Calculated Preexponential

assuming bcoll=van Der Waals radius

Calculated Preexponential assuming bcoll=covalent radius

Experimental

Å3/molec sec Å3/molec sec Preexponential

6.2 1014 2.0 1014 1.6 1014

4 1014 2.0 1014 1.1 1012

1.9 1014 7.6 1013 2.5 1013

1.25 1014 5.8 1013 1 1013

4.0 1014 2 1014 1.5 1014

Table 7.3 Preexponentials calculated from equation (7.30) for a number of reactions compared to experimental data.

Reaction

25262 HHCHCH

H CH H C2

O C H OH C H2 6 2 5

OH OH H O+O2

H O OH O2

Why Does Collision Theory Fail For Reaction 7.30?

5

Reaction 7.30 requires a special collision geometry:

(7.33)

(7.34)

•

3 2 3 3 3

3 2 3 2 2 3

CH CH CH +O: CH C HCH +•OH (7.32a)CH CH CH +O: CH CH CH +•OH (7.32b)

B

SkConfigurations = e

†

B

ΔSk configurations which lead to reactions e =

average number of configurations of the reactants

Next Few Lectures Will Cover Conventional Transition State Theory

Model reaction as motion over a potential energy surface

Use stat mech to estimate key terms6

Reaction Cordinate

Ene

rgy

ReactantsProducts

Barrier

A‡

Figure 7.5 Polanyi’s picture of excited molecules.

Objective For Today Overview of Potential Energy

Surfaces What do they look like How to interpret the plots How to interpret motion

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Figure 7.6 PE Surface For H + C2H6 →H2 + C2H5

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1 1.5 2 2.5 3

1

1.5

2

2.5

3

C-H Distance (Angstroms)

*

H-H

DIS

TAN

CE

(AN

GST

RO

MS)

C-H Distance

X

X

Y

Y

transition stateEnergy

H-H

Dist

ance

transition state

Potential Energy Surfaces

Potential energy surface is defined as the energy of the system as a function of the coordinates of all of the atoms in a reaction

Many coordinates: For H+C2H6 H2 + C2H5, 27

degrees of freedom since 9 atoms

3 translations 3 rotations, 21 others

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C-H DistanceX

Y

Saddle point

Energy

H-H

Dist

ance

Simplified Potential Energy Surfaces

Only consider bonds that break and form

Treat ligands as united atoms For A+BC AB + C, 9 degrees of

freedom since 3 atoms 3 translations 3 rotations, 3

others (AB distance, BC distance and ABC bond angle).

Textbook examples also usually assume that bond angle dependence is small

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C-H DistanceX

Y

Saddle point

Energy

H-H

Dist

ance

Simplified Potential Energy Surfaces

Simplified example: analytical PE surface

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1

11

21

31

S1 S4 S7 S10

S13

S16

S19

S22

S25

S28

S31

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

Ener

gyAB Bond Length

BC Bond Length

PE Surface

12

1

7

13

19

25

31

37

S1 S3 S5 S7 S9 S11

S13

S15

S17

S19

S21

S23

S25

S27

S29

S31

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

Spreadsheet

Numerical Values

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r2\r1 0.5 0.70 0.90 1.10 1.30 1.50 1.70 1.90 2.10 2.30 2.50 2.70 2.90 3.10 3.30 3.50 3.70 3.90 4.10 4.300.5 20.0 20.0 20.0 20.0 20.0 20.0 20.0 19.1 15.5 12.8 10.7 9.1 7.9 7.0 6.3 5.7 5.3 4.9 4.6 4.40.7 20.0 20.0 20.0 20.0 20.0 16.8 12.4 9.2 6.9 5.2 4.0 3.1 2.4 1.9 1.5 1.2 0.9 0.7 0.6 0.50.9 20.0 20.0 20.0 20.0 13.8 9.7 6.9 5.1 3.9 3.1 2.5 2.1 1.8 1.6 1.5 1.4 1.3 1.2 1.2 1.11.1 20.0 20.0 20.0 13.1 8.8 6.3 5.0 4.2 3.8 3.7 3.6 3.6 3.7 3.7 3.8 3.8 3.9 3.9 3.9 4.01.3 20.0 20.0 13.8 8.8 6.3 5.2 4.9 5.0 5.2 5.6 6.0 6.3 6.6 6.8 7.0 7.2 7.4 7.5 7.6 7.61.5 20.0 16.8 9.7 6.3 5.2 5.1 5.7 6.4 7.2 8.0 8.7 9.3 9.8 10.2 10.5 10.8 11.0 11.2 11.3 11.41.7 20.0 12.4 6.9 5.0 4.9 5.7 6.9 8.1 9.4 10.4 11.4 12.2 12.8 13.3 13.8 14.1 14.4 14.6 14.8 14.91.9 19.1 9.2 5.1 4.2 5.0 6.4 8.1 9.8 11.4 12.7 13.8 14.8 15.6 16.2 16.7 17.1 17.4 17.7 17.9 18.02.1 15.5 6.9 3.9 3.8 5.2 7.2 9.4 11.4 13.2 14.7 16.0 17.0 17.9 18.6 19.2 19.6 20.0 20.0 20.0 20.02.3 12.8 5.2 3.1 3.7 5.6 8.0 10.4 12.7 14.7 16.4 17.8 19.0 19.9 20.0 20.0 20.0 20.0 20.0 20.0 20.02.5 10.7 4.0 2.5 3.6 6.0 8.7 11.4 13.8 16.0 17.8 19.3 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.02.7 9.1 3.1 2.1 3.6 6.3 9.3 12.2 14.8 17.0 19.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.02.9 7.9 2.4 1.8 3.7 6.6 9.8 12.8 15.6 17.9 19.9 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.03.1 7.0 1.9 1.6 3.7 6.8 10.2 13.3 16.2 18.6 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.03.3 6.3 1.5 1.5 3.8 7.0 10.5 13.8 16.7 19.2 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.03.5 5.7 1.2 1.4 3.8 7.2 10.8 14.1 17.1 19.6 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.03.7 5.3 0.9 1.3 3.9 7.4 11.0 14.4 17.4 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.03.9 4.9 0.7 1.2 3.9 7.5 11.2 14.6 17.7 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.04.1 4.6 0.6 1.2 3.9 7.6 11.3 14.8 17.9 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0

Saddle Point

Spreadsheet

Top View

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A+BC AB + C1 4 7 10 13 16 19 22 25 28 31 34 37

S1

S4

S7

S10

S13

S16

S19

S22

S25

S28

S31

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

Saddle Point

AB Distance

BC D

istan

ce

Reactants

Prod

ucts

Spreadsheet

Barrierless Reaction

15

1

12

23

34

S1 S4 S7 S10

S13

S16

S19

S22

S25

S28

S31

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

Spreadsheet

Barrierless Reaction

16

1 4 7 10 13 16 19 22 25 28 31 34 37

S1

S4

S7

S10

S13

S16

S19

S22

S25

S28

S31-2

.00.

02.

04.

06.

08.

010

.012

.014

.016

.018

.020

.0

Spreadsheet

Attractive Interaction

17

1 4 7 10 13 16 19 22 25 28 31 34 37

S1

S4

S7

S10

S13

S16

S19

S22

S25

S28

S31-1

8.0

-16.

0-1

4.0

-12.

0-1

0.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

Well

Spreadsheet

PE With Van der Waals Well

18

1 4 7 10 13 16 19 22 25 28 31 34 37

S1

S4

S7

S10

S13

S16

S19

S22

S25

S28

S31-4

.0-2

.00.

02.

04.

06.

08.

010

.012

.014

.016

.018

.020

.0

SaddlePoint

Complex

Complex

Spreadsheet

PE For Series Reactions

19

1 4 7 10 13 16 19 22 25 28 31 34 37

S1

S4

S7

S10

S13

S16

S19

S22

S25

S28

S31

-12.

0-1

0.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

SaddlePoint

Intermediate

SaddlePoint

Spreadsheet

Why Do Plots Look The Way They Do?

Balance between attractive forces and Pauli repulsions

Attractive forces Van der Waals Interactions (Correlation) Bond formation

Repulsive forces Pauli repulsions (quantized electron-

electron repulsions)

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Ne-Ne Interaction

21

Ne

Ne

Ne

Ne

Separated Neons

Ne-NeCollision

Ne Ne

Anti-Bonding

Bonding

Ne-Ne Potential

22

0 1 2 3 4 5Distance Angstroms

-150

-100

-50

0

50

100E

nerg

y. K

cal/m

ol

-15

-10

-5

0

5

10

Ene

rgy,

Kca

l/mol

e

F

2

2

Ne

F-F interaction

23

F F

F F

Separated Fluorines

F2

Pure Quantum Effect

F-F Potential

24

0 1 2 3 4 5Distance Angstroms

-150

-100

-50

0

50

100

Ene

rgy.

Kca

l/mol

-15

-10

-5

0

5

10

Ene

rgy,

Kca

l/mol

e

F

2

2

Ne

Morse Potential

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V(r)=W(exp(-2x(r-ro)-2exp(-x(r-ro)))

Wherew=bond energyr=distance between atomsro=Equilibrium distanceX=range parameter

0 1 2 3 4 5Distance Angstroms

-150

-100

-50

0

50

100

Ene

rgy.

Kca

l/mol

-15

-10

-5

0

5

10

Ene

rgy,

Kca

l/mol

e

F

2

2

Ne

Cl + F2 Interaction

26

F

During Reaction

SeparatedReactants

F

F

Fluorine-Fluorine Bond

ClNon-bonding Lobe

Cl F

Fluorine-Fluorine Bond

Non-bonding Lobe

Cl + F2 Potential

27

FF

RClF

R

Energy

Interaction During H + C2H6 →CH4 + CH3

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Reactants ComeTogether,NonbondingLobe Distorts

TransitionState

H CC

Separated Reactants

H CH CH 33

Non-bonding LobesC-C bond

Rea

ctio

n P

rogr

ess

Bonds Break:

New BondsForm

Products

CH3 4CH

ReactantsBegin ToSeparate

NonbondingLobe PushsInto C-C Bond

Analytical PE Surface

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Table 7.G.1 The module used to calculate the function in equation 7.G.1

Public Function v(r1, r2, r0, a, w, vp, wa, hr) As Variantv = w * (Exp(-2 * a * (r1 - r0)) - 2 * Exp(-a * (r1 - r0)))v = v + (w + hr) * (Exp(-2 * a * (r2 - r0)) - 2 * Exp(-a * (r2 - r0)))v = v + vp * Exp(-a * (r1 + r2 - 2 * r0))v = v + wv = v + wa * Exp(-4 * a * a * ((r1 - r0) ^ 2 + (r2 - 3 * r0) ^ 2))v = v + wa * Exp(-4 * a * a * (((r1 - 3 * r0) ^ 2) + ((r2 - r0) ^ 2)))If (v > 20 + Abs(hr)) Thenv = 20 + Abs(hr)End IfEnd Function

Summary PE surface plot of energy vs internal

coordinates of reactive complex. Attractive interaction due to bonding

and Van der Waals. Repulsions due to Pauli repulsions

(quantized electron-electron repulsions).

Net yields saddle point if reaction not too exothermic.

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Question What did you learn new in this

lecture?

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